The Bernstein Function: A Unifying Framework of Nonconvex Penalization in Sparse Estimation

In this paper we study nonconvex penalization using Bernstein functions. Since the Bernstein function is concave and nonsmooth at the origin, it can induce a class of nonconvex functions for high-dimensional sparse estimation problems. We derive a threshold function based on the Bernstein penalty and give its mathematical properties in sparsity modeling. We show that a coordinate descent algorithm is especially appropriate for penalized regression problems with the Bernstein penalty. Additionally, we prove that the Bernstein function can be defined as the concave conjugate of a $\varphi$-divergence and develop a conjugate maximization algorithm for finding the sparse solution. Finally, we particularly exemplify a family of Bernstein nonconvex penalties based on a generalized Gamma measure and conduct empirical analysis for this family

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described